The Poincaré inequality is an open ended condition. (English) Zbl 1180.46025

A function \(u\) satisfies a \((1,p)\)-Poincaré inequality, \(p > 1\), if there exist \(C, \lambda \geq 1\) such that
\[ |B(x,r)|^{-1} \int_{B(x,r)} |u - u_{B(x,r)}| \,d\mu\leq Cr \bigg(|B(x,\lambda r)|^{-1} \int_{B(x,\lambda r)} g^p \,d\mu\bigg)^{1/p} \]
for every \(x \in X\) and \(r > 0\). Here, \(X\) is a metric space with a doubling Borel measure \(\mu\), \(g\) is an upper gradient of \(u\), and \(u_{B(x,r)}\) is the mean value of \(u\) in \(B(x,r)\). For \(X = \Omega\) an open set in \(\mathbb{R}^n\), and \(\mu = m\) the Lebesgue measure in \(\mathbb{R}^n\), this reduces to the ordinary Poincaré inequality of Sobolev functions, sometimes called a weak \((1,p)\)-Poincaré inequality in the case \(\lambda > 1\), see P.Hajłasz and P.Koskela [“Sobolev met Poincaré” (Mem.Am.Math.Soc.688) (2000; Zbl 0954.46022)] for various aspects of Poincaré and Sobolev inequalities.
The authors show that, if \(X\) is a complete metric space, then there exists \(\varepsilon > 0\) such that a \((1,q)\)-Poincaré inequality holds for \(q >p-\varepsilon\) whenever a \((1,p)\)-Poincaré inequality holds, i.e., that the Poincaré inequality is an open ended property. The result has an important consequence for \(p\)-admissible weights in \(\mathbb{R}^n\): they display the same open ended property as Muckenhoupt’s \(A_p\) weights. It also shows that various definitions for Sobolev spaces in metric spaces, due to J.Cheeger [Geom.Funct.Anal.9, No.3, 428–517 (1999; Zbl 0942.58018)], P.Hajłasz [Potential Anal.5, No.4, 403–415 (1996; Zbl 0859.46022)] and N.Shanmugalingam [Rev.Mat.Iberoam.16, No.2, 243–279 (2000; Zbl 0974.46038)], coincide under the above assumptions. The proof makes use of careful estimates of the maximal function of \(u - u_{B(x,r)}\).


46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
43A85 Harmonic analysis on homogeneous spaces
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